Some Homology and Cohomology Theories for a Metric Space

Robert Hardt (Rice University)

BLAINEFEST

October 27, 2012 Coauthors

Thierry De Pauw (Paris VII) -H., Rectifiable and Flat G Chains in a Metric Space Amer.J.Math.2011

De Pauw, -H., Washek Pfeffer (UC Davis, Emeritus) Homology of Normal Chains and Cohomology of Charges In preparation.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 2 / 27 One can first try to have a simple (small) theory adapted to the spaces being considered. For example, * For M triangulated, use simplicial theory. * For M a smooth manifold and for real coefficients, use differential forms and De Rham theory. * For M semi-algebraic, use semi-algebraic chains, etc.

A Vague Problem from Algebraic Topology

m For a given homology class α ∈ Hm(M) or cohomology class β ∈ H (M), find a “special” representative cycle or cocycle.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 3 / 27 For example, * For M triangulated, use simplicial theory. * For M a smooth manifold and for real coefficients, use differential forms and De Rham theory. * For M semi-algebraic, use semi-algebraic chains, etc.

A Vague Problem from Algebraic Topology

m For a given homology class α ∈ Hm(M) or cohomology class β ∈ H (M), find a “special” representative cycle or cocycle. One can first try to have a simple (small) theory adapted to the spaces being considered.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 3 / 27 * For M a smooth manifold and for real coefficients, use differential forms and De Rham theory. * For M semi-algebraic, use semi-algebraic chains, etc.

A Vague Problem from Algebraic Topology

m For a given homology class α ∈ Hm(M) or cohomology class β ∈ H (M), find a “special” representative cycle or cocycle. One can first try to have a simple (small) theory adapted to the spaces being considered. For example, * For M triangulated, use simplicial theory.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 3 / 27 * For M semi-algebraic, use semi-algebraic chains, etc.

A Vague Problem from Algebraic Topology

m For a given homology class α ∈ Hm(M) or cohomology class β ∈ H (M), find a “special” representative cycle or cocycle. One can first try to have a simple (small) theory adapted to the spaces being considered. For example, * For M triangulated, use simplicial theory. * For M a smooth manifold and for real coefficients, use differential forms and De Rham theory.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 3 / 27 A Vague Problem from Algebraic Topology

m For a given homology class α ∈ Hm(M) or cohomology class β ∈ H (M), find a “special” representative cycle or cocycle. One can first try to have a simple (small) theory adapted to the spaces being considered. For example, * For M triangulated, use simplicial theory. * For M a smooth manifold and for real coefficients, use differential forms and De Rham theory. * For M semi-algebraic, use semi-algebraic chains, etc.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 3 / 27 * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure.

Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 Minimizers One can also look for special representatives using a variational property. * In Hodge theory for a Riemannian manifold, harmonic forms minimize an L2 norm in a De Rham class. * Also some geodesics provide length minimizing cycles in a one dimensional integral homology class.

This example and the obstruction (Thom) to any smooth manifold representatives of various higher dimensional integral homology classes led to the question of existence of mass-minimizing representing cycles. In 1960, H. Federer and W. Fleming studied not only the (absolute) Plateau problem of finding a mass-minimizers of general dimension with a given boundary. They also considered the corresponding problem of minimizing mass in a given homology class. This required the chains of the homology theory to have a suitable notion of mass and a suitable topology to give limits of mass-minimizing sequences. The chains should include oriented finite volume submanifolds and should, in general have some geometrc structure. Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 4 / 27 Remark. ELNR’s include compact smooth submanifolds and polygons, but not pieces of algebraic subvarieties with cusps. What are rectifiable chains?

Rectifiable Chains

Theorem. (H.Federer - W.Fleming, 1959) Integer-multiplicity rectifiable chains give the ordinary integral homology for pairs of compact Euclidean Lipschitz neighborhood retracts (ELNR). Homology classes of such pairs contain mass-minimizing rectifiable chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 5 / 27 Remark. ELNR’s include compact smooth submanifolds and polygons, but not pieces of algebraic subvarieties with cusps. What are rectifiable chains?

Rectifiable Chains

Theorem. (H.Federer - W.Fleming, 1959) Integer-multiplicity rectifiable chains give the ordinary integral homology for pairs of compact Euclidean Lipschitz neighborhood retracts (ELNR). Homology classes of such pairs contain mass-minimizing rectifiable chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 5 / 27 What are rectifiable chains?

Rectifiable Chains

Theorem. (H.Federer - W.Fleming, 1959) Integer-multiplicity rectifiable chains give the ordinary integral homology for pairs of compact Euclidean Lipschitz neighborhood retracts (ELNR). Homology classes of such pairs contain mass-minimizing rectifiable chains.

Remark. ELNR’s include compact smooth submanifolds and polygons, but not pieces of algebraic subvarieties with cusps.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 5 / 27 Rectifiable Chains

Theorem. (H.Federer - W.Fleming, 1959) Integer-multiplicity rectifiable chains give the ordinary integral homology for pairs of compact Euclidean Lipschitz neighborhood retracts (ELNR). Homology classes of such pairs contain mass-minimizing rectifiable chains.

Remark. ELNR’s include compact smooth submanifolds and polygons, but not pieces of algebraic subvarieties with cusps. What are rectifiable chains?

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 5 / 27 m Parameterization Theorem. There exist disjoint compact Ai ⊂ R and ∞ m an injective map α : A = ∪i=1Ai → M such that H [M \ α(A)] = 0, −1 √ Lip α ≤ 1, and Lip(α  Ai ) ≤ 2 m.

Rectifiable Sets A subset M of a metric space X is Hm rectifiable if Hm(X \ f (E)) = 0 for m some Lebesgue measurable E ⊂ R and Lipschitz f : E → M. Here Hm is m dimensional Hausdorff measure.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 6 / 27 m Parameterization Theorem. There exist disjoint compact Ai ⊂ R and ∞ m an injective map α : A = ∪i=1Ai → M such that H [M \ α(A)] = 0, −1 √ Lip α ≤ 1, and Lip(α  Ai ) ≤ 2 m.

Rectifiable Sets A subset M of a metric space X is Hm rectifiable if Hm(X \ f (E)) = 0 for m some Lebesgue measurable E ⊂ R and Lipschitz f : E → M. Here Hm is m dimensional Hausdorff measure.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 6 / 27 Rectifiable Sets A subset M of a metric space X is Hm rectifiable if Hm(X \ f (E)) = 0 for m some Lebesgue measurable E ⊂ R and Lipschitz f : E → M. Here Hm is m dimensional Hausdorff measure.

m Parameterization Theorem. There exist disjoint compact Ai ⊂ R and ∞ m an injective map α : A = ∪i=1Ai → M such that H [M \ α(A)] = 0, −1 √ Lip α ≤ 1, and Lip(α  Ai ) ≤ 2 m.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 6 / 27 Rectifiable Sets A subset M of a metric space X is Hm rectifiable if Hm(X \ f (E)) = 0 for m some Lebesgue measurable E ⊂ R and Lipschitz f : E → M. Here Hm is m dimensional Hausdorff measure.

m Parameterization Theorem. There exist disjoint compact Ai ⊂ R and ∞ m an injective map α : A = ∪i=1Ai → M such that H [M \ α(A)] = 0, −1 √ Lip α ≤ 1, and Lip(α  Ai ) ≤ 2 m.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 6 / 27 We make the identification [[α, A, g]] = [[β, B, h]] if Z Z |g ◦ α−1| dHm = 0 = |h ◦ β−1| dHm α(A)\β(B) β(B)\α(A)

and, Hm a.e. on α−1[α(A) ∩ β(B)],

g = [sgn det D(β−1) ◦ α] h ◦ β−1 ◦ α
.

Rm(X ; G) = {m dimensional rectifiable G chains T in X }.

Rectifiable G Chains Let (G, k k) be a complete normed abelian group. We get a rectifiable G chain simply by adding a density function g ∈ L1(A, G) to our parameterization.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 7 / 27 Rm(X ; G) = {m dimensional rectifiable G chains T in X }.

Rectifiable G Chains Let (G, k k) be a complete normed abelian group. We get a rectifiable G chain simply by adding a density function g ∈ L1(A, G) to our parameterization. We make the identification [[α, A, g]] = [[β, B, h]] if Z Z |g ◦ α−1| dHm = 0 = |h ◦ β−1| dHm α(A)\β(B) β(B)\α(A)

and, Hm a.e. on α−1[α(A) ∩ β(B)],

g = [sgn det D(β−1) ◦ α] h ◦ β−1 ◦ α
.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 7 / 27 Rm(X ; G) = {m dimensional rectifiable G chains T in X }.

Rectifiable G Chains Let (G, k k) be a complete normed abelian group. We get a rectifiable G chain simply by adding a density function g ∈ L1(A, G) to our parameterization. We make the identification [[α, A, g]] = [[β, B, h]] if Z Z |g ◦ α−1| dHm = 0 = |h ◦ β−1| dHm α(A)\β(B) β(B)\α(A)

and, Hm a.e. on α−1[α(A) ∩ β(B)],

g = [sgn det D(β−1) ◦ α] h ◦ β−1 ◦ α
.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 7 / 27 Rectifiable G Chains Let (G, k k) be a complete normed abelian group. We get a rectifiable G chain simply by adding a density function g ∈ L1(A, G) to our parameterization. We make the identification [[α, A, g]] = [[β, B, h]] if Z Z |g ◦ α−1| dHm = 0 = |h ◦ β−1| dHm α(A)\β(B) β(B)\α(A)

and, Hm a.e. on α−1[α(A) ∩ β(B)],

g = [sgn det D(β−1) ◦ α] h ◦ β−1 ◦ α
.

Rm(X ; G) = {m dimensional rectifiable G chains T in X }.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 7 / 27 In fact, for any dense subset D of X and point x0 in X , let

ι : X → `∞(D) = {bounded functions on D} ,

x ∈ X 7→ dist (·, x) − dist (·, x0) . Thus, identifying X with ι(X ), we may now think of X itself as being a subset of Y = `∞(D). In particular, the standard space `∞ of bounded sequences essentially contains any separable metric metric space.

Embedding Remark

Kuratowski made the beautiful observation that Any metric space X admits distance-preserving map into a Banach space.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 8 / 27 Thus, identifying X with ι(X ), we may now think of X itself as being a subset of Y = `∞(D). In particular, the standard space `∞ of bounded sequences essentially contains any separable metric metric space.

Embedding Remark

Kuratowski made the beautiful observation that Any metric space X admits distance-preserving map into a Banach space.

In fact, for any dense subset D of X and point x0 in X , let

ι : X → `∞(D) = {bounded functions on D} ,

x ∈ X 7→ dist (·, x) − dist (·, x0) .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 8 / 27 In particular, the standard space `∞ of bounded sequences essentially contains any separable metric metric space.

Embedding Remark

Kuratowski made the beautiful observation that Any metric space X admits distance-preserving map into a Banach space.

In fact, for any dense subset D of X and point x0 in X , let

ι : X → `∞(D) = {bounded functions on D} ,

x ∈ X 7→ dist (·, x) − dist (·, x0) . Thus, identifying X with ι(X ), we may now think of X itself as being a subset of Y = `∞(D).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 8 / 27 Embedding Remark

Kuratowski made the beautiful observation that Any metric space X admits distance-preserving map into a Banach space.

In fact, for any dense subset D of X and point x0 in X , let

ι : X → `∞(D) = {bounded functions on D} ,

x ∈ X 7→ dist (·, x) − dist (·, x0) . Thus, identifying X with ι(X ), we may now think of X itself as being a subset of Y = `∞(D). In particular, the standard space `∞ of bounded sequences essentially contains any separable metric metric space.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 8 / 27 R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 Mass, Polyhedral, and Lipschitz Chains R −1 m Mass M(T ) = M[[α, A, g]] = α(A) kg ◦ α k dH . Suppose Y = `∞(D) contains X as before. PI A polyhedral G chain in Y is simply a finite sum P = i=1 [[γi , ∆i , gi ]] m where γi : R → Y is affine, ∆i is an m simplex, and gi is constant on ∆i .

A Lipschitz chain in Y is defined similarly except that now the γi are arbitrary Lipschitz maps into Y .

Let Pm(Y ; G) and Lm(Y ; G) denote the groups of m dimensional polyhedral and Lipschitz chains. Then: The rectifiable chains Rm(Y , G) is the mass completion of Lm(Y , G). Polyhedral and Lipschitz chains have easily defined boundary operations, but these are not mass continuous. As the Koch snowflake in the plane shows, the boundary of a rectifiable chain is not expected to be rectifiable in general. So defining it requires completion of Lipschitz chains with respect to a weaker norm.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 9 / 27 Whitney defined the flat norm, which we adapt. For a Lipschitz chain T ∈ Lm(Y ; G), let

F(T ) = inf{M(S) + M(T − ∂S): S ∈ Lm+1(Y , G)} .

Then the flat norm F([[1/i]] − [[0]]) ≤ 1/i → 0 because [[1/i]] − [[0]] = ∂[[0, 1/i]].

Since F is a norm on Lm(Y ; G), we can define the group of flat chains Fm(Y ; G) is the F completion of Lm(Y ; G). (or alternately of Pm(Y ; G)) The flat continuity of ∂ on Lipschitz chains gives a well-defined boundary operator on the flat chains Fm(Y ; G). Since F ≤ M, a rectifiable chain T ∈ Rm(Y ; G) is flat and so now has a well-defined boundary ∂T ∈ Fm−1(Y ; G).

Flat Norm and Flat Chains Note that in the space R the points 1/i approach the point 0, but the corresponding 0 dimensional chains [[1/i]] do not approach [[0]] in mass norm because M([[1/i]] − [[0]]) = 2.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 10 / 27 Since F is a norm on Lm(Y ; G), we can define the group of flat chains Fm(Y ; G) is the F completion of Lm(Y ; G). (or alternately of Pm(Y ; G)) The flat continuity of ∂ on Lipschitz chains gives a well-defined boundary operator on the flat chains Fm(Y ; G). Since F ≤ M, a rectifiable chain T ∈ Rm(Y ; G) is flat and so now has a well-defined boundary ∂T ∈ Fm−1(Y ; G).

Flat Norm and Flat Chains Note that in the space R the points 1/i approach the point 0, but the corresponding 0 dimensional chains [[1/i]] do not approach [[0]] in mass norm because M([[1/i]] − [[0]]) = 2. Whitney defined the flat norm, which we adapt. For a Lipschitz chain T ∈ Lm(Y ; G), let

F(T ) = inf{M(S) + M(T − ∂S): S ∈ Lm+1(Y , G)} .

Then the flat norm F([[1/i]] − [[0]]) ≤ 1/i → 0 because [[1/i]] − [[0]] = ∂[[0, 1/i]].

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 10 / 27 The flat continuity of ∂ on Lipschitz chains gives a well-defined boundary operator on the flat chains Fm(Y ; G). Since F ≤ M, a rectifiable chain T ∈ Rm(Y ; G) is flat and so now has a well-defined boundary ∂T ∈ Fm−1(Y ; G).

Flat Norm and Flat Chains Note that in the space R the points 1/i approach the point 0, but the corresponding 0 dimensional chains [[1/i]] do not approach [[0]] in mass norm because M([[1/i]] − [[0]]) = 2. Whitney defined the flat norm, which we adapt. For a Lipschitz chain T ∈ Lm(Y ; G), let

F(T ) = inf{M(S) + M(T − ∂S): S ∈ Lm+1(Y , G)} .

Then the flat norm F([[1/i]] − [[0]]) ≤ 1/i → 0 because [[1/i]] − [[0]] = ∂[[0, 1/i]].

Since F is a norm on Lm(Y ; G), we can define the group of flat chains Fm(Y ; G) is the F completion of Lm(Y ; G). (or alternately of Pm(Y ; G))

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 10 / 27 Since F ≤ M, a rectifiable chain T ∈ Rm(Y ; G) is flat and so now has a well-defined boundary ∂T ∈ Fm−1(Y ; G).

Flat Norm and Flat Chains Note that in the space R the points 1/i approach the point 0, but the corresponding 0 dimensional chains [[1/i]] do not approach [[0]] in mass norm because M([[1/i]] − [[0]]) = 2. Whitney defined the flat norm, which we adapt. For a Lipschitz chain T ∈ Lm(Y ; G), let

F(T ) = inf{M(S) + M(T − ∂S): S ∈ Lm+1(Y , G)} .

Then the flat norm F([[1/i]] − [[0]]) ≤ 1/i → 0 because [[1/i]] − [[0]] = ∂[[0, 1/i]].

Since F is a norm on Lm(Y ; G), we can define the group of flat chains Fm(Y ; G) is the F completion of Lm(Y ; G). (or alternately of Pm(Y ; G)) The flat continuity of ∂ on Lipschitz chains gives a well-defined boundary operator on the flat chains Fm(Y ; G).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 10 / 27 Flat Norm and Flat Chains Note that in the space R the points 1/i approach the point 0, but the corresponding 0 dimensional chains [[1/i]] do not approach [[0]] in mass norm because M([[1/i]] − [[0]]) = 2. Whitney defined the flat norm, which we adapt. For a Lipschitz chain T ∈ Lm(Y ; G), let

F(T ) = inf{M(S) + M(T − ∂S): S ∈ Lm+1(Y , G)} .

Then the flat norm F([[1/i]] − [[0]]) ≤ 1/i → 0 because [[1/i]] − [[0]] = ∂[[0, 1/i]].

Since F is a norm on Lm(Y ; G), we can define the group of flat chains Fm(Y ; G) is the F completion of Lm(Y ; G). (or alternately of Pm(Y ; G)) The flat continuity of ∂ on Lipschitz chains gives a well-defined boundary operator on the flat chains Fm(Y ; G). Since F ≤ M, a rectifiable chain T ∈ Rm(Y ; G) is flat and so now has a well-defined boundary ∂T ∈ Fm−1(Y ; G).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 10 / 27 We now have the closed subgroups of cycles F Zm (X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , ∂T = 0} for m ≥ 1 , F Z0 (X ; G) = {T ∈ F0(Y ; G): sptT ⊂ X , χ(T ) = 0} , P∞ P∞ where χ( i=1 gi [[xi ]]) = i=1 gi , and the flat chains homology groups F F Hm(X ; G) = Zm (X ; G)/{∂S : S ∈ Fm+1(Y ; G), sptT ⊂ X }. Let

Nm(X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , M(T ) + M(∂T ) < ∞} denote the subgroup of normal chains. Then working with either rectifiable chains having rectifiable boundaries or with normal chains, one can similarly define rectifiable chains homology R Hm (X ; G) and normal chains homology

Hm(X ; G) .

Flat, Rectifiable, and Normal Chains Homology

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 11 / 27 Let

Nm(X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , M(T ) + M(∂T ) < ∞} denote the subgroup of normal chains. Then working with either rectifiable chains having rectifiable boundaries or with normal chains, one can similarly define rectifiable chains homology R Hm (X ; G) and normal chains homology

Hm(X ; G) .

Flat, Rectifiable, and Normal Chains Homology We now have the closed subgroups of cycles F Zm (X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , ∂T = 0} for m ≥ 1 , F Z0 (X ; G) = {T ∈ F0(Y ; G): sptT ⊂ X , χ(T ) = 0} , P∞ P∞ where χ( i=1 gi [[xi ]]) = i=1 gi , and the flat chains homology groups F F Hm(X ; G) = Zm (X ; G)/{∂S : S ∈ Fm+1(Y ; G), sptT ⊂ X }.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 11 / 27 Then working with either rectifiable chains having rectifiable boundaries or with normal chains, one can similarly define rectifiable chains homology R Hm (X ; G) and normal chains homology

Hm(X ; G) .

Flat, Rectifiable, and Normal Chains Homology We now have the closed subgroups of cycles F Zm (X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , ∂T = 0} for m ≥ 1 , F Z0 (X ; G) = {T ∈ F0(Y ; G): sptT ⊂ X , χ(T ) = 0} , P∞ P∞ where χ( i=1 gi [[xi ]]) = i=1 gi , and the flat chains homology groups F F Hm(X ; G) = Zm (X ; G)/{∂S : S ∈ Fm+1(Y ; G), sptT ⊂ X }. Let

Nm(X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , M(T ) + M(∂T ) < ∞} denote the subgroup of normal chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 11 / 27 Flat, Rectifiable, and Normal Chains Homology We now have the closed subgroups of cycles F Zm (X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , ∂T = 0} for m ≥ 1 , F Z0 (X ; G) = {T ∈ F0(Y ; G): sptT ⊂ X , χ(T ) = 0} , P∞ P∞ where χ( i=1 gi [[xi ]]) = i=1 gi , and the flat chains homology groups F F Hm(X ; G) = Zm (X ; G)/{∂S : S ∈ Fm+1(Y ; G), sptT ⊂ X }. Let

Nm(X ; G) = {T ∈ Fm(Y ; G): sptT ⊂ X , M(T ) + M(∂T ) < ∞} denote the subgroup of normal chains. Then working with either rectifiable chains having rectifiable boundaries or with normal chains, one can similarly define rectifiable chains homology R Hm (X ; G) and normal chains homology

Hm(X ; G) .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 11 / 27 Let’s look at a few examples relevant to mass-minimizing G chains.

An Example

2 For X being the standard fractal boundary of the Koch snowflake in R ,

R F H1(X ; Z) = 0 , H1 (X ; Z) = 0 , and H1 (X ; Z) = Z

because X supports no nonzero rectifiable or finite mass one dimensional flat chains though X itself is the support of a nonzero infinite mass flat cycle (that bounds in Y ).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 12 / 27 An Example

2 For X being the standard fractal boundary of the Koch snowflake in R ,

R F H1(X ; Z) = 0 , H1 (X ; Z) = 0 , and H1 (X ; Z) = Z

because X supports no nonzero rectifiable or finite mass one dimensional flat chains though X itself is the support of a nonzero infinite mass flat cycle (that bounds in Y ).

Let’s look at a few examples relevant to mass-minimizing G chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 12 / 27 1966 W. Fleming used chains with coefficients in a finite abelian group. 3 Example 1. For a minimal Mobius band, A in R viewed as a Z/2Z chain, ∂A is a circle.

Example 2. B is three (similarly-oriented) semi-circles bounding A which is three half-disks. Here ∂B = 0 and ∂A = B as Z/3Z chains.

Very Short History

n 1960 H. Federer-W. Fleming used chains with R or Z coefficients in R . Here the chains are currents, i.e. linear functionals on differential forms.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 13 / 27 3 Example 1. For a minimal Mobius band, A in R viewed as a Z/2Z chain, ∂A is a circle.

Example 2. B is three (similarly-oriented) semi-circles bounding A which is three half-disks. Here ∂B = 0 and ∂A = B as Z/3Z chains.

Very Short History

n 1960 H. Federer-W. Fleming used chains with R or Z coefficients in R . Here the chains are currents, i.e. linear functionals on differential forms. 1966 W. Fleming used chains with coefficients in a finite abelian group.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 13 / 27 Example 2. B is three (similarly-oriented) semi-circles bounding A which is three half-disks. Here ∂B = 0 and ∂A = B as Z/3Z chains.

Very Short History

n 1960 H. Federer-W. Fleming used chains with R or Z coefficients in R . Here the chains are currents, i.e. linear functionals on differential forms. 1966 W. Fleming used chains with coefficients in a finite abelian group. 3 Example 1. For a minimal Mobius band, A in R viewed as a Z/2Z chain, ∂A is a circle.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 13 / 27 Very Short History

n 1960 H. Federer-W. Fleming used chains with R or Z coefficients in R . Here the chains are currents, i.e. linear functionals on differential forms. 1966 W. Fleming used chains with coefficients in a finite abelian group. 3 Example 1. For a minimal Mobius band, A in R viewed as a Z/2Z chain, ∂A is a circle.

Example 2. B is three (similarly-oriented) semi-circles bounding A which is three half-disks. Here ∂B = 0 and ∂A = B as Z/3Z chains.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 13 / 27 2000 L. Ambrosio and B. Kirchheim Chains are newly defined currents in a metric space ( which have R or Z coefficients).

2002 Jerrard, 2003 H.-DePauw, 2005 T. Adams, 2007 S. Wenger, 2007 U. Lang, 2009 Ambrosio-Wenger, 2009 Ambrosio-Katz, 2009 M. Snipes, 2010 C. Reidwig, 2011 Wenger

An important property relevant to the existence of mass-minimizing chains is a suitable compactness theorem. Our version is the following:

Short History Cont’d 1999 B. White treated general normed abelian coefficient groups with new proofs. White’s and Fleming’s chains are obtained by completing groups of elementary chains with respect to suitable metrics.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 14 / 27 2002 Jerrard, 2003 H.-DePauw, 2005 T. Adams, 2007 S. Wenger, 2007 U. Lang, 2009 Ambrosio-Wenger, 2009 Ambrosio-Katz, 2009 M. Snipes, 2010 C. Reidwig, 2011 Wenger

An important property relevant to the existence of mass-minimizing chains is a suitable compactness theorem. Our version is the following:

Short History Cont’d 1999 B. White treated general normed abelian coefficient groups with new proofs. White’s and Fleming’s chains are obtained by completing groups of elementary chains with respect to suitable metrics. 2000 L. Ambrosio and B. Kirchheim Chains are newly defined currents in a metric space ( which have R or Z coefficients).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 14 / 27 An important property relevant to the existence of mass-minimizing chains is a suitable compactness theorem. Our version is the following:

Short History Cont’d 1999 B. White treated general normed abelian coefficient groups with new proofs. White’s and Fleming’s chains are obtained by completing groups of elementary chains with respect to suitable metrics. 2000 L. Ambrosio and B. Kirchheim Chains are newly defined currents in a metric space ( which have R or Z coefficients).

2002 Jerrard, 2003 H.-DePauw, 2005 T. Adams, 2007 S. Wenger, 2007 U. Lang, 2009 Ambrosio-Wenger, 2009 Ambrosio-Katz, 2009 M. Snipes, 2010 C. Reidwig, 2011 Wenger

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 14 / 27 Short History Cont’d 1999 B. White treated general normed abelian coefficient groups with new proofs. White’s and Fleming’s chains are obtained by completing groups of elementary chains with respect to suitable metrics. 2000 L. Ambrosio and B. Kirchheim Chains are newly defined currents in a metric space ( which have R or Z coefficients).

2002 Jerrard, 2003 H.-DePauw, 2005 T. Adams, 2007 S. Wenger, 2007 U. Lang, 2009 Ambrosio-Wenger, 2009 Ambrosio-Katz, 2009 M. Snipes, 2010 C. Reidwig, 2011 Wenger

An important property relevant to the existence of mass-minimizing chains is a suitable compactness theorem. Our version is the following:

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 14 / 27 (II) KR ∩ Rm(X ; G) is F compact in case G contains no nonconstant Lipschitz curve*.

*, discovered by B. White, is true for G = Z or Z/jZ but not (R, | |).

The rectifiability in (B) give the desired geometric character to the Plateau problem solutions. While this rectifiabiity is not true for G = R with the usual absolute value norm | |, it is true for another norm on R:

Compactness Theorem

Theorem. [DHP] Suppose X is a compact metric space and G is a complete normed group with closed balls being compact. For R > 0,

(I) KR = {T ∈ Fm(X ; G): M(T ) + M(∂T ) ≤ R} is F compact.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 15 / 27 The rectifiability in (B) give the desired geometric character to the Plateau problem solutions. While this rectifiabiity is not true for G = R with the usual absolute value norm | |, it is true for another norm on R:

Compactness Theorem

Theorem. [DHP] Suppose X is a compact metric space and G is a complete normed group with closed balls being compact. For R > 0,

(I) KR = {T ∈ Fm(X ; G): M(T ) + M(∂T ) ≤ R} is F compact.

(II) KR ∩ Rm(X ; G) is F compact in case G contains no nonconstant Lipschitz curve*.

*, discovered by B. White, is true for G = Z or Z/jZ but not (R, | |).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 15 / 27 Compactness Theorem

Theorem. [DHP] Suppose X is a compact metric space and G is a complete normed group with closed balls being compact. For R > 0,

(I) KR = {T ∈ Fm(X ; G): M(T ) + M(∂T ) ≤ R} is F compact.

(II) KR ∩ Rm(X ; G) is F compact in case G contains no nonconstant Lipschitz curve*.

*, discovered by B. White, is true for G = Z or Z/jZ but not (R, | |).

The rectifiability in (B) give the desired geometric character to the Plateau problem solutions. While this rectifiabiity is not true for G = R with the usual absolute value norm | |, it is true for another norm on R:

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 15 / 27 n We may connect two probability measures µ, ν in R by choosing n T ∈ F1(R , G) with ∂T = µ − ν. α For 0 < α < 1, we define the norm krkα = |r| for r ∈ R. Then (R, k · kα) does satisfy condition * . Also “merging” paths in T may reduce the corresponding mass Mα(T ).

Example.

√ √ M 1 (T ) = 1 · (6 + 6) > 1 · 4 2 + 2 · 4 = M 1 (S). 2 2

Shared Transport Paths (mail, biology, economics, etc.)

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 16 / 27 α For 0 < α < 1, we define the norm krkα = |r| for r ∈ R. Then (R, k · kα) does satisfy condition * . Also “merging” paths in T may reduce the corresponding mass Mα(T ).

Example.

√ √ M 1 (T ) = 1 · (6 + 6) > 1 · 4 2 + 2 · 4 = M 1 (S). 2 2

Shared Transport Paths (mail, biology, economics, etc.) n We may connect two probability measures µ, ν in R by choosing n T ∈ F1(R , G) with ∂T = µ − ν.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 16 / 27 Example.

√ √ M 1 (T ) = 1 · (6 + 6) > 1 · 4 2 + 2 · 4 = M 1 (S). 2 2

Shared Transport Paths (mail, biology, economics, etc.) n We may connect two probability measures µ, ν in R by choosing n T ∈ F1(R , G) with ∂T = µ − ν. α For 0 < α < 1, we define the norm krkα = |r| for r ∈ R. Then (R, k · kα) does satisfy condition * . Also “merging” paths in T may reduce the corresponding mass Mα(T ).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 16 / 27 Shared Transport Paths (mail, biology, economics, etc.) n We may connect two probability measures µ, ν in R by choosing n T ∈ F1(R , G) with ∂T = µ − ν. α For 0 < α < 1, we define the norm krkα = |r| for r ∈ R. Then (R, k · kα) does satisfy condition * . Also “merging” paths in T may reduce the corresponding mass Mα(T ).

Example.

√ √ M 1 (T ) = 1 · (6 + 6) > 1 · 4 2 + 2 · 4 = M 1 (S). 2 2

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 16 / 27 Regularity Theorem.(Q. Xia) spt T \ (spt µ ∪ spt ν) is locally a polygon.

Higher Dimensions.(H.–De Pauw, In progress) For m ≥ 1, dim (spt T \spt ∂T ) ≤ m − 1 for any Mα minimizing T ∈ Rm(X , Z).

Mα Minimizers

n Corollary.(Q. Xia) There exists a Mα minimizing T ∈ R1(R , R) with ∂T = µ − ν.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 17 / 27 Higher Dimensions.(H.–De Pauw, In progress) For m ≥ 1, dim (spt T \spt ∂T ) ≤ m − 1 for any Mα minimizing T ∈ Rm(X , Z).

Mα Minimizers

n Corollary.(Q. Xia) There exists a Mα minimizing T ∈ R1(R , R) with ∂T = µ − ν.

Regularity Theorem.(Q. Xia) spt T \ (spt µ ∪ spt ν) is locally a polygon.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 17 / 27 Mα Minimizers

n Corollary.(Q. Xia) There exists a Mα minimizing T ∈ R1(R , R) with ∂T = µ − ν.

Regularity Theorem.(Q. Xia) spt T \ (spt µ ∪ spt ν) is locally a polygon.

Higher Dimensions.(H.–De Pauw, In progress) For m ≥ 1, dim (spt T \spt ∂T ) ≤ m − 1 for any Mα minimizing T ∈ Rm(X , Z).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 17 / 27 Proof of (I)

ˆ KR is F complete by the lower semicontinuity of M. So we need only show that KR is also totally bounded. For this, it suffices to find, for each ε > 0, a compact subset Cε of Fm(Y ; G) so that

KR ⊂ {T ∈ Fm(Y ; G): distF (T , Cε) < 2εR} .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 18 / 27 First, note that

I  X Cε = gi Qi : Qi = m cube of a size ε subdivision, i=1

I X m Qi ∩ p(X ) 6= ∅, and kgi kε ≤ cR i=1

is F compact. Now, for any T ∈ KR , an affine homotopy shows that F(p#T − T ) ≤ εR. Next the Deformation Theorem implies that F(p#T − Q) ≤ εR for some Q ∈ Cε. So distF (T , Cε) < 2εR.

Continuation of Proof of (I) By the MAP (Metric Approximation Property) of Y = `∞(D) there is a Lipschitz 1 linear projection p of Y onto some finite n dimensional W ⊂ Y so that kp(x) − xk < ε for all x in the compact set X . W is n equivalent to R (with bounds only depending on X and ε). So we assume n W = R and use the Deformation Theorem of B. White.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 19 / 27 Now, for any T ∈ KR , an affine homotopy shows that F(p#T − T ) ≤ εR. Next the Deformation Theorem implies that F(p#T − Q) ≤ εR for some Q ∈ Cε. So distF (T , Cε) < 2εR.

Continuation of Proof of (I) By the MAP (Metric Approximation Property) of Y = `∞(D) there is a Lipschitz 1 linear projection p of Y onto some finite n dimensional W ⊂ Y so that kp(x) − xk < ε for all x in the compact set X . W is n equivalent to R (with bounds only depending on X and ε). So we assume n W = R and use the Deformation Theorem of B. White. First, note that

I  X Cε = gi Qi : Qi = m cube of a size ε subdivision, i=1

I X m Qi ∩ p(X ) 6= ∅, and kgi kε ≤ cR i=1 is F compact.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 19 / 27 Continuation of Proof of (I) By the MAP (Metric Approximation Property) of Y = `∞(D) there is a Lipschitz 1 linear projection p of Y onto some finite n dimensional W ⊂ Y so that kp(x) − xk < ε for all x in the compact set X . W is n equivalent to R (with bounds only depending on X and ε). So we assume n W = R and use the Deformation Theorem of B. White. First, note that

I  X Cε = gi Qi : Qi = m cube of a size ε subdivision, i=1

I X m Qi ∩ p(X ) 6= ∅, and kgi kε ≤ cR i=1

is F compact. Now, for any T ∈ KR , an affine homotopy shows that F(p#T − T ) ≤ εR. Next the Deformation Theorem implies that F(p#T − Q) ≤ εR for some Q ∈ Cε. So distF (T , Cε) < 2εR.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 19 / 27 Then we apply the lemma with µ = |νT | where νT is the nonatomic part of T . Fot nonzero µ, we get, in the group G, the nonconstant continuous  −1  curve γ(t) = νT f [0, t) of finite length ≤ M(T ), contradicting (*).

Proof of (II), m = 0

For the case m = 0, we follow the argument of White and note that T is a G valued Borel measure, which we wish to show is purely atomic. First we verify the general

Lemma. For any positive Borel measure µ without atoms on X , there exists a µ measurable function f : X → [0, 1] so that µ[f −1{t}] = 0 for every t ∈ [0, 1].

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 20 / 27 Proof of (II), m = 0

For the case m = 0, we follow the argument of White and note that T is a G valued Borel measure, which we wish to show is purely atomic. First we verify the general

Lemma. For any positive Borel measure µ without atoms on X , there exists a µ measurable function f : X → [0, 1] so that µ[f −1{t}] = 0 for every t ∈ [0, 1].

Then we apply the lemma with µ = |νT | where νT is the nonatomic part of T . Fot nonzero µ, we get, in the group G, the nonconstant continuous  −1  curve γ(t) = νT f [0, t) of finite length ≤ M(T ), contradicting (*).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 20 / 27 For the case m > 0 we generalize Jerrard’s observation showing that, for m any Lipschitz map f : X → R , the slice function

m hT , f , ·i ∈ BV (R , R0(X ; G)) .

By an argument of Ambrosio-Kirchheim, this may be approximated by a Lipschitz function, leading to the rectifiability of T .

Proof of (II), m > 0

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 21 / 27 By an argument of Ambrosio-Kirchheim, this may be approximated by a Lipschitz function, leading to the rectifiability of T .

Proof of (II), m > 0

For the case m > 0 we generalize Jerrard’s observation showing that, for m any Lipschitz map f : X → R , the slice function

m hT , f , ·i ∈ BV (R , R0(X ; G)) .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 21 / 27 Proof of (II), m > 0

For the case m > 0 we generalize Jerrard’s observation showing that, for m any Lipschitz map f : X → R , the slice function

m hT , f , ·i ∈ BV (R , R0(X ; G)) .

By an argument of Ambrosio-Kirchheim, this may be approximated by a Lipschitz function, leading to the rectifiability of T .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 21 / 27 Hm(X ) = Zm(X )/{∂S : S ∈ Nm+1(X )} .

Real Normal Chains and Dual Cochains For simplicity, we will, for the rest of the lecture, assume that X is a compact metric space, G is the coefficeint group R with the standard norm | | , and drop the R symbol. Thus we have the vector space

Nm(X ) = {T ∈ Fm(X , R): M(T ) + M(∂T ) < ∞}

of normal R chains in X as well as the closed subspaces of cycles

Zm(X ) = {T ∈ Nm(X ): ∂T = 0} for m ≥ 1 ,

Z0(X ) = {T ∈ N0(X ): T (1) = 0} .

Here a T ∈ N0(X ) corresponds to a signed Borel measure on X , T (1) denotes its total integral over X , and

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 22 / 27 Real Normal Chains and Dual Cochains For simplicity, we will, for the rest of the lecture, assume that X is a compact metric space, G is the coefficeint group R with the standard norm | | , and drop the R symbol. Thus we have the vector space

Nm(X ) = {T ∈ Fm(X , R): M(T ) + M(∂T ) < ∞}

of normal R chains in X as well as the closed subspaces of cycles

Zm(X ) = {T ∈ Nm(X ): ∂T = 0} for m ≥ 1 ,

Z0(X ) = {T ∈ N0(X ): T (1) = 0} .

Here a T ∈ N0(X ) corresponds to a signed Borel measure on X , T (1) denotes its total integral over X , and

Hm(X ) = Zm(X )/{∂S : S ∈ Nm+1(X )} .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 22 / 27 D. Sullivan, J. Heinenon, and S. Keith used such forms to study local bilipschitz equivalence to Euclidean space. M. Snipes (2009) gave a generalization of Wolfe’s theorem to Banach spaces with a new notion of partial form. Charges, which act on normal chains, were defined by De Pauw to study R R solutions of div v = F by using the terms of ∂Ω v · ν = Ω F as functionals of the set Ω of finite perimeter. n De Pauw, Moonens, Pfeffer (2009) showed that charges in R correspond to ω + dη for some continuous ω, η.

Cochains

n ∗ Whitney also studied the dual space Fm(R ; R) of flat cochains, and his student J. Wolfe (1957) showed that any flat cochain comes from bounded Borel m form ω where dω is a bounded Borel m + 1 forms. This n means α(T ) = T (ω) for T ∈ Fm(R ; R).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 23 / 27 M. Snipes (2009) gave a generalization of Wolfe’s theorem to Banach spaces with a new notion of partial form. Charges, which act on normal chains, were defined by De Pauw to study R R solutions of div v = F by using the terms of ∂Ω v · ν = Ω F as functionals of the set Ω of finite perimeter. n De Pauw, Moonens, Pfeffer (2009) showed that charges in R correspond to ω + dη for some continuous ω, η.

Cochains

n ∗ Whitney also studied the dual space Fm(R ; R) of flat cochains, and his student J. Wolfe (1957) showed that any flat cochain comes from bounded Borel m form ω where dω is a bounded Borel m + 1 forms. This n means α(T ) = T (ω) for T ∈ Fm(R ; R). D. Sullivan, J. Heinenon, and S. Keith used such forms to study local bilipschitz equivalence to Euclidean space.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 23 / 27 Charges, which act on normal chains, were defined by De Pauw to study R R solutions of div v = F by using the terms of ∂Ω v · ν = Ω F as functionals of the set Ω of finite perimeter. n De Pauw, Moonens, Pfeffer (2009) showed that charges in R correspond to ω + dη for some continuous ω, η.

Cochains

n ∗ Whitney also studied the dual space Fm(R ; R) of flat cochains, and his student J. Wolfe (1957) showed that any flat cochain comes from bounded Borel m form ω where dω is a bounded Borel m + 1 forms. This n means α(T ) = T (ω) for T ∈ Fm(R ; R). D. Sullivan, J. Heinenon, and S. Keith used such forms to study local bilipschitz equivalence to Euclidean space. M. Snipes (2009) gave a generalization of Wolfe’s theorem to Banach spaces with a new notion of partial form.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 23 / 27 Cochains

n ∗ Whitney also studied the dual space Fm(R ; R) of flat cochains, and his student J. Wolfe (1957) showed that any flat cochain comes from bounded Borel m form ω where dω is a bounded Borel m + 1 forms. This n means α(T ) = T (ω) for T ∈ Fm(R ; R). D. Sullivan, J. Heinenon, and S. Keith used such forms to study local bilipschitz equivalence to Euclidean space. M. Snipes (2009) gave a generalization of Wolfe’s theorem to Banach spaces with a new notion of partial form. Charges, which act on normal chains, were defined by De Pauw to study R R solutions of div v = F by using the terms of ∂Ω v · ν = Ω F as functionals of the set Ω of finite perimeter. n De Pauw, Moonens, Pfeffer (2009) showed that charges in R correspond to ω + dη for some continuous ω, η.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 23 / 27 A charge is a continuous linear α :(Nm(X ), TN) → R. Let

CHm(X ) = {m dimensional charges in X } .

We have the continuous operators

δ : CHm(X ) → CHm+1(X ) , (δα)(S) = α(∂S)

# m m # φ : CH (Y ) → CH (X ), (φ α)(T ) = α(φ#T ) for Lipschitz φ : X → Y .

Charges

The localized topology TN on Nm(X ) has the property that ˆ ˆ Tj → T in TN ⇐⇒ F(Tj − T ) → 0 and sup M(Tj ) + M(∂Tj ) < ∞ . j

(For noncompact X , one should add ∪j spt Tj ⊂ single compact set.)

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 24 / 27 Charges

The localized topology TN on Nm(X ) has the property that ˆ ˆ Tj → T in TN ⇐⇒ F(Tj − T ) → 0 and sup M(Tj ) + M(∂Tj ) < ∞ . j

(For noncompact X , one should add ∪j spt Tj ⊂ single compact set.)

A charge is a continuous linear α :(Nm(X ), TN) → R. Let

CHm(X ) = {m dimensional charges in X } .

We have the continuous operators

δ : CHm(X ) → CHm+1(X ) , (δα)(S) = α(∂S)

# m m # φ : CH (Y ) → CH (X ), (φ α)(T ) = α(φ#T ) for Lipschitz φ : X → Y .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 24 / 27 ∞ 3 Example. X = ∪i=1Xi where Xi are embedded curves in R joining points i ai to 0, disjoint away from 0, and with length(Xi ) = 2 . Then X is P∞ −i Lipschitz path connected, but T = [[0]] − i=1 2 [[ai ]] has χ(T ) = 0 although T bounds no one chain of finite mass in X . So (B) does not imply (A) in general. Theorem. (C) implies (A) if X satisfies the linear isoperimetric condition

c0(X ) = inf{M(S)/M(∂S): S ∈ N1(X ) } < ∞ .

Definition. X is m bounded ⇐⇒ M(S) ≤ cm(X )M(∂S) for all S ∈ Nm+1(X ). This linearly isoperimetric condition has been studied by many people (Gromov,... ,Wenger).

0 Vanishing of H0(X ), H (X ) Theorem. Consider the following three conditions. (A) H0(X ) = 0. (B) X is Lipschitz path connected. (C) H0(X ) = 0. Then (A) implies (B) and (B) implies (C).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 25 / 27 Theorem. (C) implies (A) if X satisfies the linear isoperimetric condition

c0(X ) = inf{M(S)/M(∂S): S ∈ N1(X ) } < ∞ .

Definition. X is m bounded ⇐⇒ M(S) ≤ cm(X )M(∂S) for all S ∈ Nm+1(X ). This linearly isoperimetric condition has been studied by many people (Gromov,... ,Wenger).

0 Vanishing of H0(X ), H (X ) Theorem. Consider the following three conditions. (A) H0(X ) = 0. (B) X is Lipschitz path connected. (C) H0(X ) = 0. Then (A) implies (B) and (B) implies (C). ∞ 3 Example. X = ∪i=1Xi where Xi are embedded curves in R joining points i ai to 0, disjoint away from 0, and with length(Xi ) = 2 . Then X is P∞ −i Lipschitz path connected, but T = [[0]] − i=1 2 [[ai ]] has χ(T ) = 0 although T bounds no one chain of finite mass in X . So (B) does not imply (A) in general.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 25 / 27 Definition. X is m bounded ⇐⇒ M(S) ≤ cm(X )M(∂S) for all S ∈ Nm+1(X ). This linearly isoperimetric condition has been studied by many people (Gromov,... ,Wenger).

0 Vanishing of H0(X ), H (X ) Theorem. Consider the following three conditions. (A) H0(X ) = 0. (B) X is Lipschitz path connected. (C) H0(X ) = 0. Then (A) implies (B) and (B) implies (C). ∞ 3 Example. X = ∪i=1Xi where Xi are embedded curves in R joining points i ai to 0, disjoint away from 0, and with length(Xi ) = 2 . Then X is P∞ −i Lipschitz path connected, but T = [[0]] − i=1 2 [[ai ]] has χ(T ) = 0 although T bounds no one chain of finite mass in X . So (B) does not imply (A) in general. Theorem. (C) implies (A) if X satisfies the linear isoperimetric condition

c0(X ) = inf{M(S)/M(∂S): S ∈ N1(X ) } < ∞ .

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 25 / 27 0 Vanishing of H0(X ), H (X ) Theorem. Consider the following three conditions. (A) H0(X ) = 0. (B) X is Lipschitz path connected. (C) H0(X ) = 0. Then (A) implies (B) and (B) implies (C). ∞ 3 Example. X = ∪i=1Xi where Xi are embedded curves in R joining points i ai to 0, disjoint away from 0, and with length(Xi ) = 2 . Then X is P∞ −i Lipschitz path connected, but T = [[0]] − i=1 2 [[ai ]] has χ(T ) = 0 although T bounds no one chain of finite mass in X . So (B) does not imply (A) in general. Theorem. (C) implies (A) if X satisfies the linear isoperimetric condition

c0(X ) = inf{M(S)/M(∂S): S ∈ N1(X ) } < ∞ .

Definition. X is m bounded ⇐⇒ M(S) ≤ cm(X )M(∂S) for all S ∈ Nm+1(X ). This linearly isoperimetric condition has been studied by many people (Gromov,... ,Wenger). Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 25 / 27 Using the corresponding charge cohomology groups m H (X ) = ker δm/im δm−1, we have the

Duality Theorem On the category of pairs of compact metric spaces satisfying all m boundedness conditions (also relative versions ), m H and Hm satisfy the Eilenberg-Steenrod axioms, and the two functors m ∗ H and Hm are naturally equivalent.

In 1974 Federer proved a duality theory using real flat chains and flat cochains for the category of Euclidean Lipschitz neighborhood retracts. Our goal with normal chain homology and charge cohomology is to understand metric properties of more general spaces such as varieties, fractals, or Gromov-Hausdorff limits of manifolds.

Duality

Theorem. X is m bounded ⇐⇒ {∂S : S ∈ Nm+1(X )} is TN closed. ⇒ {δβ : β ∈ CHm(X )} is closed in CHm+1(X ).

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 26 / 27 Duality Theorem On the category of pairs of compact metric spaces satisfying all m boundedness conditions (also relative versions ), m H and Hm satisfy the Eilenberg-Steenrod axioms, and the two functors m ∗ H and Hm are naturally equivalent.

In 1974 Federer proved a duality theory using real flat chains and flat cochains for the category of Euclidean Lipschitz neighborhood retracts. Our goal with normal chain homology and charge cohomology is to understand metric properties of more general spaces such as varieties, fractals, or Gromov-Hausdorff limits of manifolds.

Duality

Theorem. X is m bounded ⇐⇒ {∂S : S ∈ Nm+1(X )} is TN closed. ⇒ {δβ : β ∈ CHm(X )} is closed in CHm+1(X ).

Using the corresponding charge cohomology groups m H (X ) = ker δm/im δm−1, we have the

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 26 / 27 In 1974 Federer proved a duality theory using real flat chains and flat cochains for the category of Euclidean Lipschitz neighborhood retracts. Our goal with normal chain homology and charge cohomology is to understand metric properties of more general spaces such as varieties, fractals, or Gromov-Hausdorff limits of manifolds.

Duality

Theorem. X is m bounded ⇐⇒ {∂S : S ∈ Nm+1(X )} is TN closed. ⇒ {δβ : β ∈ CHm(X )} is closed in CHm+1(X ).

Using the corresponding charge cohomology groups m H (X ) = ker δm/im δm−1, we have the

Duality Theorem On the category of pairs of compact metric spaces satisfying all m boundedness conditions (also relative versions ), m H and Hm satisfy the Eilenberg-Steenrod axioms, and the two functors m ∗ H and Hm are naturally equivalent.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 26 / 27 Duality

Theorem. X is m bounded ⇐⇒ {∂S : S ∈ Nm+1(X )} is TN closed. ⇒ {δβ : β ∈ CHm(X )} is closed in CHm+1(X ).

Using the corresponding charge cohomology groups m H (X ) = ker δm/im δm−1, we have the

Duality Theorem On the category of pairs of compact metric spaces satisfying all m boundedness conditions (also relative versions ), m H and Hm satisfy the Eilenberg-Steenrod axioms, and the two functors m ∗ H and Hm are naturally equivalent.

In 1974 Federer proved a duality theory using real flat chains and flat cochains for the category of Euclidean Lipschitz neighborhood retracts. Our goal with normal chain homology and charge cohomology is to understand metric properties of more general spaces such as varieties, fractals, or Gromov-Hausdorff limits of manifolds.

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 26 / 27 HAPPY BIRTHDAY BLAINE !

Questions

(1) Determine when various specific spaces are m bounded.

(2) Interpret Hm(X ) and Hm(X ) (as Banach spaces) for m ≥ 1.

(3) Is there special “regularity” for mass-minimizers?

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 27 / 27 Questions

(1) Determine when various specific spaces are m bounded.

(2) Interpret Hm(X ) and Hm(X ) (as Banach spaces) for m ≥ 1.

(3) Is there special “regularity” for mass-minimizers?

HAPPY BIRTHDAY BLAINE !

Robert Hardt (Rice University) (BLAINEFEST)Some Homology and Cohomology Theories for a Metric SpaceOctober 27, 2012 27 / 27